Nonlocal Mean Field Schrödinger Bridge with Learned Interactions

2026-06-02

AI Summary

This work addresses the computational intractability of mean-field Schrödinger bridge problems, where nonlocal interactions induce a quadratic scaling of complexity with particle count, hindering scalability to large systems. To overcome this, the authors propose an efficient solver leveraging a neural network surrogate model to approximate the nonlocal interaction term—a first in this context—and introduce a four-stage alternating optimization algorithm. This approach reduces the per-step inference complexity from O(N²) to O(N). Theoretically, they establish a Grönwall-type stability bound to quantify the impact of surrogate approximation error on trajectory accuracy. Empirically, the method reproduces high-fidelity trajectories in navigation and opinion dynamics tasks while substantially accelerating training, thereby demonstrating both accuracy and scalability.

Abstract

The Schrödinger Bridge Problem constructs a stochastic process that connects an initial distribution to a terminal distribution with minimum energy. This work considers its mean-field extension, the Mean-Field Schrödinger Bridge, for interacting particle systems. With nonlocal interactions, evaluating the resulting particle-dependent distributional terms can scale quadratically with the population size, which makes large-scale problems intractable. We address this bottleneck by approximating the nonlocal interactions with neural network surrogates. The resulting four-stage alternating algorithm reduces the per-step cost from quadratic to linear in the population size at inference. We also derive Grönwall-type stability bounds that show how surrogate errors propagate to the generated trajectories. In numerical experiments on navigation and opinion-dynamics tasks, the proposed method reproduces trajectories obtained with analytical evaluation and reduces training time.

Problem

• Nonlocal Interactions
• Mean-Field Schrödinger Bridge
• Scalability
• Particle Systems
• Computational Complexity

Innovation

• Nonlocal Mean Field
• Schrödinger Bridge
• Neural Network Surrogate
• Scalable Inference
• Stability Analysis